Affine combinations in affine schemes

arXiv:1508.04322v1 [math.AG] 18 Aug 2015

Affine combinations in affine schemes Anders Kock

Introduction The notion of “neighbour points” in algebraic geometry is a geometric rendering of the notion of nilpotent elements in commutative rings, and was developed since the time of Study, Kähler, Hjelmslev, and notably in French algebraic geometry (Grothendieck, Weil et al.) since the 1950s. They introduced it via what they call the first neighbourhood of the diagonal. In [3], [4] and [8] the neighbour notion was considered on an axiomatic basis, essentially for finite dimensional manifolds; one of the aims was to describe a combinatorial theory of differential forms. In the specific context of algebraic geometry, such theory of differential forms was developed in [1], where it applies not only to manifolds, but to arbitrary schemes. One aspect, present in [4] and [8], but not in [1], is the possibility of forming affine combinations of finite sets of mutual (1st order) neighbour points. The present note completes this aspect, by giving the construction of such affine combinations, at least in the category of affine schemes (the dual of the category of finitely presented commutative rings or k-algebras). The interest in having the possibility of such affine combinations is documented in several places in [8], and is in [4] the basis for constructing, for any manifold, a simplicial object, whose cochain complex is the deRham complex of the manifold. From a more philosophical viewpoint, one may say that the possibility of having affine combinations, for sets of mutual neighbour points, expresses in a concrete way the idea that spaces are “infinitesimally like affine spaces”.

1

1 Neighbour maps between algebras Let k be a commutative ring. Consider commutative k-algebras B and C and two k-algebra maps f and g : B → C. We say that they are neighbours, or more completely, (first order) infinitesimal neighbours, if ( f (a) − g(a)) · ( f (b) − g(b)) = 0 for all a, b ∈ B,

(1)

or equivalently, if f (a) · g(b) + g(a) · f (b) = f (a · b) + g(a · b) for all a, b ∈ B.

(2)

(Note that this latter formulation makes no use of “minus”.) When this holds, we write f ∼ g (or more completely, f ∼1 g). The relation ∼ is a reflexive and symmetric relation (but not transitive). If the element 2 ∈ k is invertible, a third equivalent formulation of f ∼ g goes ( f (a) − g(a))2 = 0 for all a ∈ B.

(3)

For, it is clear that (1) implies (3). Conversely, assume (3), and let a, b ∈ B be arbitrary, and apply (3) to the element a + b. Then by assumption, and using that f and g are algebra maps1 , 0 = ( f (a + b) − g(a + b))2 = [( f (a) − g(a)) + ( f (b) − g(b))]2 = ( f (a) − g(a))2 + ( f (b) − g(b))2 − 2( f (a) − g(a)) · ( f (b) − g(b)). The two first terms are 0 by assumption, hence so is the third. Now divide by 2. Note that if C has no zero-divisors, then f ∼ g is equivalent to f = g. It is clear that the relation ∼ is stable under precomposition: if h : B′ → B and f ∼ g : B → C, then f ◦ h ∼ g ◦ h : B′ → C,

(4)

and, using that h is an algebra map, it is also stable under postcomposition: if h : C → C′ and f ∼ g : B → C, then h ◦ f ∼ h ◦ g : B → C′ .

(5)

1 “algebra” means throughout “commutative k-algebra”, and similarly for algebra maps. When we say “linear map”, we mean k-linear. By ⊗, we mean ⊗k .

2

Also, if h : B′ → B is a surjective algebra map, precomposition by h not only preserves the neighbour relation, it also reflects it, in the following sense f ◦ h ∼ g ◦ h implies f ∼ g.

(6)

This is immediate from (1); the a and b occurring there is of the form h(a′ ) and h(b′ ) for suitable a′ and b′ in B′ , by surjectivity of h. An alternative “element-free” formulation of the neighbour relation (Proposition 1.2 below) comes from a standard piece of commutative algebra. Recall that for commutative k-algebras A and B, the tensor product A ⊗ B carries structure of commutative k-algebra (A ⊗ B is in fact a coproduct of A and B); the multiplication map m : B ⊗ B → B is a k-algebra homomorphism; so the kernel is an ideal J ⊆ B ⊗ B. The following is a classical description of the ideal J ⊆ B ⊗ B; we include it for completeness. Proposition 1.1 The kernel J of m : B ⊗ B → B is generated by the expressions 1 ⊗ b − b ⊗ 1, for b ∈ B. Hence the ideal J 2 is generated by the expressions (1 ⊗ a − a ⊗ 1) · (1 ⊗ b − b ⊗ 1) (or equivalently, by the expressions 1 ⊗ ab + ab ⊗ 1 − a ⊗ b − b ⊗ a). Proof. It is clear that 1 ⊗ b − b ⊗ 1 is in J. Conversely, assume that ∑i ai ⊗ bi is in J, i.e. that ∑i ai · bi = 0. Rewrite the ith term ai ⊗ bi as follows: ai ⊗ bi = ai bi ⊗ 1 + (ai ⊗ 1) · (1 ⊗ bi − bi ⊗ 1) and sum over i; since ∑i ai bi = 0, we are left with ∑i (ai ⊗ 1) · (1 ⊗ bi − bi ⊗ 1), which belongs to the B ⊗ B-module generated by elements of the form 1 ⊗ b − b ⊗ 1. – The second assertion follows, since ab ⊗ 1 + 1 ⊗ ab − a ⊗ b − b ⊗ a is the product of the two generators 1 ⊗ a − a ⊗ 1 and 1 ⊗ b − b ⊗ 1. (Note that the proof gave a slightly stronger result, namely that J is generated already as a Bmodule, by the elements 1 ⊗ b − b ⊗ 1, via the algebra map i0 : B → B ⊗ B, where i0 (b) = b ⊗ 1). From the second assertion in this Proposition immediately follows that f ∼ g iff { f , g} : B ⊗ B → C factors across the quotient map B ⊗ B → (B ⊗ B)/J 2 (where { f , g} : B ⊗ B → C denotes the map given by a ⊗ b 7→ f (a) · g(b)); equivalently: Proposition 1.2 For f , g : B → C, we have f ∼ g if and only if { f , g} : B ⊗ B → C annihilates J 2 . 3

The two natural inclusion maps i0 and i1 : B → B ⊗ B (given by b 7→ b ⊗ 1 and b 7→ 1 ⊗ b, respectively) are not in general neighbours, but when postcomposed with π : B ⊗ B → (B ⊗ B)/J 2 , they are:

π ◦ i0 ∼ π ◦ i1 , and this is in fact the universal pair of neighbour algebra maps with domain B.

2 Neighbours for polynomial algebras We consider the polynomial algebra B := k[X1, . . . , Xn ]. Identifying B ⊗ B with k[Y1 , . . . ,Yn , Z1 , . . . , Zn ], the multiplication map m is the algebra map given by Yi 7→ Xi and Zi 7→ Xi , so it is clear that the kernel J of m contains the n elements Zi −Yi . The following Proposition should be classical: Proposition 2.1 The ideal J ⊆ B ⊗ B, for B = k[X1, . . . , Xn ], is generated (as a B ⊗ B-module) by the n elements Zi −Yi . Proof. From Proposition 1.1, we know that J is generated by elements P(Z) − P(Y ), for P ∈ k[X] (where X denotes X1 , . . . , Xn , and similarly for Y and Z). So it suffices to prove that P(Z) − P(Y ) is of the form n

∑ (Zi −Yi )Qi(Y , Z).

i=1

This is done by induction in n. For n = 1, it suffices, by linearity, to prove this fact for each monomial X s . And this follows from the identity Z s −Y s = (Z −Y ) · (Z s−1 + Z s−2Y + . . . + ZY s−2 +Y s−1 )

(7)

(for s ≥ 1; for s = 0, we get 0). For the induction step: Write P(X) as a sum of increasing powers of X1 , P(X1, X2 , . . .) = P0 (X2, . . .) + X1 P1 (X2, . . .) + X12P2 (X2 , . . .). Apply the induction hypothesis to the first term. The remaining terms are of the form X1s Qs (X2 , . . .) with s ≥ 1; then the difference to be considered is Y1s Qs (Y2 , . . .) − Z1s Qs (Z2 , . . .) 4

which we may write as Y1s (Qs (Y2 , . . .) − Qs (Z2 , . . .)) + Qs (Z2 , . . .)(Y1s − Z1s ). The first term in this sum is taken care of by the induction hypothesis, the second term uses the identity (7) which shows that this term is in the ideal generated by (Z1 −Y1 ). From this follows immediately Proposition 2.2 The ideal J 2 ⊆ B ⊗ B, for B = k[X1, . . . , Xn] is generated (as a B ⊗ B-module) by the elements (Zi − Yi )(Z j − Y j ) (for i, j = 1, . . . , n) (identifying B ⊗ B with k[Y1 , . . .,Yn , Z1 , . . . , Zn ]). (The algebra (B ⊗ B)/J 2 is the algebra representing the affine scheme “first neighbourhood of the diagonal” for the affine scheme represented by B, alluded to in the introduction.) Algebra maps a : k[X1 , . . ., Xn ] → C are completely given by an n-tuple of elements ai := a(Xi ) ∈ C (i = 1, . . . , n). Let b : k[X1, . . . , Xn ] → C be similarly given by the n-tuple bi ∈ C. The decision when a ∼ b can be expressed equationally in terms of these two n-tuples of elements in C, i.e. as a purely equationally described condition on elements (a1 , . . ., an , b1 , . . . , bn ) ∈ C2n : Proposition 2.3 Consider two algebra maps a and b : k[X1, . . . , Xn] → C. Let ai := a(Xi ) and bi := b(Xi). Then we have a ∼ b if and only if (bi − ai ) · (b j − a j ) = 0

(8)

for all i, j = 1, . . . , n. Proof. We have that a ∼ b iff the algebra map {a, b} annihilates the ideal J 2 for the algebra k[X1, . . . , Xn ]; and this in turn is equivalent to that it annihilates the set of generators for J 2 described in the Proposition 2.2). But {a, b}((Zi −Yi ) · (Z j − Y j )) = (bi − ai ) · (b j − a j ), and then the result is immediate. We therefore also say that the pair of n-tuples of elements in C   a1 . . . an b1 . . . bn are neighbours if (8) holds. 5

For brevity, we call an n-tuple (c1 , . . . , cn ) of elements in Cn a vector, and denote it c . Thus a vector (c1 , . . ., cn ) is neighbour of the “zero” vector 0 = (0, . . ., 0) iff ci · c j = 0 for all i and j. Remark. Even when 2 ∈ k is invertible, one cannot conclude that a ∼ b follows from (bi − ai )2 = 0 for all i = 1, . . ., n. For, consider C := k[ε1 , ε2 ] = k[ε ] ⊗ k[ε ] (where k[ε ] is the “ring of dual numbers over k”, so ε 2 = 0). Then the pair of n-tuples (n = 2 here) given by (a1 , a2 ) = (ε1 , ε2) and (b1 , b2 ) := (0, 0) has (ai − bi )2 = εi2 = 0 for i = 1, 2, but (a1 − b1 ) · (a2 − b2 ) = ε1 · ε2 , which is not 0 in C. We already have the notion of when two algebra maps f and g : B → C are neighbours, or infinitesimal neighbours. We also say that the pair ( f , g) form an infinitesimal 1-simplex (with f and g as vertices). Also, we have the derived (8) notion of when two vectors in Cn are neighbours, or form an infinitesimal 1simplex. This terminology is suited for being generalized to defining the notion of infinitesimal p-simplex of algebra maps B → C, or of infinitesimal p-simplex of vectors in Cn (for p = 1, 2, . . .). Proposition 2.3 generalizes immediately to infinitesimal p-simplices (where the Proposition is the special case of p = 1): Proposition 2.4 Consider p + 1 algebra maps ai : k[X1 , . . . , Xn] → C (for i = 0, . . ., p), and let ai j ∈ C be ai (X j ), for j = 1, . . . n. Then the ai form an infinitesimal p-simplex iff for all i, i′ = 0, . . . p and j, j′ = 1, . . . , n (ai j − ai′ j ) · (ai j′ − ai′ j′ ) = 0.

(9)

3 Affine combinations of mutual neighbours Let C be a k-algebra. An affine combination in a C-module means here a linear combination in the module, with coefficients from C, and where the sum of the coefficients is 1 . We consider in particular the C-module Link (B,C) of k-linear maps B → C, where B is another k-algebra. Linear combinations of algebra maps are linear, but may fail to preserve the multiplicative structure and 1. However Theorem 3.1 Let f0 , . . ., f p be a p + 1-tuple of mutual neighbour algebra maps B → C, and let t0 , . . .,t p be elements of C with t0 + . . . + t p = 1. Then the affine combination p

∑ ti · f i : B → C

i=0

6

is an algebra map. Composing with a map h : C → C′ preserves the affine combination. Proof. Since the sum is a k-linear map, it suffices to prove that it preserves the multiplicative structure. It clearly preserves 1. To prove that it preserves products a · b, we should compare (∑ ti fi (a)) · (∑ t j f j (b)) = ∑ tit j fi (a) · f j (b) i

j

i, j

with ∑ ti fi (a · b). Now use that ∑ j t j = 1; then ∑ ti fi (a · b) may be rewritten as

∑ tit j fi(a · b). ij

Compare the two displayed double sums: the terms with i = j match since each fi preserves multiplication. Consider a pair of indices i 6= j; the terms with index i j and ji from the first sum contribute tit j times fi (a) · f j (b) + f j (a) · fi(b),

(10)

and the terms terms with index i j and ji from the second sum contribute tit j times fi (a · b) + f j (a · b),

(11)

and the two displayed contributions are equal, since fi ∼ f j (use the formulation (2)). The last assertion is obvious from the construction. Theorem 3.2 Let Let f0 , . . . , f p be a p + 1-tuple of mutual neighbour algebra maps B → C. Then any two affine combinations (with coefficients from C) of these maps are neighbours. Proof. Let ∑i ti fi and ∑ j s j f j be two such affine combinations. To prove that they are neighbours means (using (2)) to prove that for all a and b in B, (∑ ti fi (a)) · (∑ s j f j (b)) + (∑ s j f j (a)) · (∑ ti fi (b)) i

equals

j

j

∑ ti fi(a · b) + ∑ s j f j (a · b). i

j

7

i

The first of these expressions equals

∑ tis j fi(a) · f j(b) + ∑ tis j f j (a) · fi(b) = ∑ tis j [ fi(a) · f j(b) + f j (a) · fi(b)] ij

ij

ij

For the second expression, we use ∑ j s j = 1 and ∑i ti = 1, to rewrite it as the left hand expression in

∑ tis j fi(a · b) + ∑ tis j f j (a · b) = ∑ tis j [ fi(a · b) + f j(a · b)]. ij

ij

ij

For each i j, the two square bracket expression match by (2), since fi ∼ f j . Combining these two results, we have Theorem 3.3 Let f0 , . . ., f p be a p + 1-tuple of mutual neighbour algebra maps B → C. Then in the C-module of C-linear maps B → C, the affine subspace AffC ( f0 , . . ., f p ) of affine combinations (with coefficients from C) of the fi s consists of algebra maps, and they are mutual neighbours. (2)

2 in the p + 1In [1], they describe an ideal J0p . It is the sum of ideals Jrs fold tensor product B ⊗ . . . ⊗ B, where Jrs is the ideal generated by is (b) − ir (b) for (2) b ∈ B and r < s. We shall here denote it just J for brevity; it has the property that the p + 1 inclusions B → B ⊗ . . .⊗ B) become mutual neighbours, when composed (2) with the quotient map π : B ⊗ . . . ⊗ B → (B ⊗ . . . ⊗ B)/J , and this is in fact the universal p + 1 tuple of mutual neighbour maps with domain B. We may, for any given k-algebra B, encode the construction of Theorem 3.1 into one single canonical map which does not mention any individual B → C, by using the universal p+1-tuple, and the generic p+1 tuple of coefficients with sum 1, meaning (X0 , X1, . . . , X p) ∈ k[X1 , . . ., X p ] (where X0 denotes 1 − (X1 + . . . + X p); namely as a k-algebra map

B → (B⊗k+1 /J

(2)

) ⊗ k[X1 , . . ., X p].

(12)

For, by the Yoneda Lemma, this is equivalent to giving a (set theoretical) map, natural in C, hom((B⊗p+1 /J

(2)

) ⊗ k[X1, . . . , X p],C) → hom(B,C),

(where hom denotes the set of k-algebra maps). An element on the left hand side is given by a p +1-tuple of mutual neighbouring algebra maps fi : B → C, together 8

p

with a p-tuple (t1 , . . .,t p) of elements in C. With t0 := 1 − ∑1 ti , such data produce an element ∑0p ti · fi in hom(B,C), by Theorem 3.1,and the construction is natural in C by the last assertion in the Theorem. (2) The affine scheme defined by the algebra B⊗p+1 /J is (essentially) called (p) ∆B in [1], and, (in axiomatic context, and for manifolds, in a suitable sense), the corresponding object is called M[p] in [4] and M(1,1,...,1) in [3] I. 18 (for suitable M).

4 Affine combinations in a k-algebra C The constructions and results of the previous Section concerning infinitesimal psimplices of algebra maps B → C, specializes (by taking B = k[X1, . . . , Xn], as in Section 2) to infinitesimal p-simplices of vectors in Cn ; such a p-simplex is conveniently exhibited in a (p + 1) × n matrix with entries ai j from C:   a01 . . . a0n  a11 . . . a1n     .. ..  .  . .  a p1 . . . a pn We may of course form affine (or even linear) combinations, with coefficients from C, of the rows of this matrix, whether or not the rows are mutual neighbours. But the same affine combination of the corresponding algebra maps is in general only a k-linear map, not an algebra map. However, if the rows are mutual neighbours in Cn , and hence the corresponding algebra maps are mutual neighbouring algebra maps k[X1 , . . . , Xn] → C, we have, by Theorem 3.1 that the affine combinations of the rows of the matrix corresponds to the similar affine combination of the algebra maps. For, it suffices to check their equality on the Xi s, since the Xi s generate k[X1 , . . ., Xn ] as an algebra. Therefore, the Theorems 3.2 and 3.3 immediately translate into theorems about p + 1-tuples of mutual neighbouring n-tuples of elements in the algebra C; recall that such a p + 1-tuple may be identified with the rows of a (p + 1) × n matrix with entries from C, satisfying the equations (9). Theorem 4.1 Let the rows of a (p + 1) × n matrix with entries from C be mutual neighbours. Then any two affine combinations (with coefficients from C) of these rows are neighbours. The set of all such affine combinations form an affine subspace of the C-module Cn . 9

Let us consider in particular the case where the 0th row of a (p + 1) × n matrix is the zero vector (0, . . ., 0). Then the following is an elementary calculation: Proposition 4.2 Consider a (p + 1) × n matrix {ai j } as above, but with a0 j = 0 for j = 1, . . .n. Then the rows form an infinitesimal p-simplex iff the conjunction of ai j · ai′ j′ + ai′ j · ai j′ = 0 for all i, i′ = 1, . . . p, j, j′ = 1, . . .n. (13) hold. and ai j · ai j′ = 0 for all i = 1, . . . , p, j = 1, . . . n

(14)

If 2 is invertible in C, the equations (14) follow from (13). Proof. The last assertion follows by putting i = i′ in (13), and dividing by 2. Assume that the rows of the matrix form an infinitesimal p-simplex. Then (14) follows from ai ∼ 0. The equation which asserts that ai ∼ ai′ (for i, i′ = 1, . . . , p) is (ai j − ai′ j ) · (ai j′ − ai′ j′ ) = 0 for all j, j′ = 1, . . .n. Multiplying out gives four terms, two of which vanish by virtue of (14), and the two remaining add up to (minus) the sum on the left of (13). For the converse implication, (14) give that the last p rows are ∼ 0; and (14) and (13) jointly give that ai ∼ ai′ , by essentially the same calculation which we have already made. When 0 is one of the vectors in a p + 1-tuple, any linear combination of the remaining p vectors has the same value as a certain affine combination of all p + 1 vectors, since the coefficient for 0 may be chosen arbitrarily without changing the value of the linear combination. Therefore the results on affine combinations of the rows in the (p+1)×n matrix with 0 as top row immediately translate to results about linear combinations of the remaining rows, i.e. they translate into results about p × n matrices, satisfying the equations (13) and (14); even the equations (13) suffice, if 2 is invertible. In this form, the results were obtained in the preprint [6], and are stated here for completeness. We assume that 2 ∈ k is invertible. We use the notation from [3] I.16 and I. 18, where set of p × n matrices {ai j } ˜ satisfying (13) was denoted D(p, n) ⊆ C p·n (we there consider algebras C over ˜ 2) consists of matrices of the form k = Q, so (14) follows). In particular D(2,   a11 a12 with a11 · a22 + a12 · a21 = 0. a21 a22 Note that the determinant of such a matrix is 2 times the product of the diagonal ˜ 2) is stable under transposition of matrices. entries. And also note that D(2, 10

˜ The notation D(p, n) may be consistently augmented to the case where p = 1; ˜ n) if it is neighbour of 0 ∈ Cn , i.e. if a j · a j′ = 0 for all we say (a1 , . . . , an ) ∈ D(1, ˜ n) is also denoted D(n), and D(1) is denoted D.) j, j′ = 1, . . . n. (In [3], D(1, ˜ n) precisely when all its 2 × 2 It is clear that a p × n matrix belongs to D(p, sub-matrices do; this is just a reflection of the fact that the defining equations (13) only involve two row indices and two column indices at a time. From the ˜ 2) therefore follows that transposition p×n matrices transposition stability of D(2, ˜ n) into D(n, ˜ p). takes D(p, ˜ n) is a neighbour of 0 ∈ Cn . Note that each of the rows of a matrix in D(p, The results about affine combinations now get the following formulations in ˜ terms of linear combinations of the rows of matrices in D(p, n): ˜ n). Let a (p +1) ×n matrix X ′ be obtained Theorem 4.3 Given a matrix X ∈ D(p, by adjoining to X a row which is a linear combination of the rows of X . Then X ′ ˜ + 1, n). is in D(p

5 Geometric meaning This Section contains essentially only reformulations of the previous material into geometric terms, and thereby it also contains some motivation of the notions. In any category E , a map f : I → M may be thought of either as an I-parametrized family of elements (or points) of M, or as an M-valued function on I. This terminology is convenient in particular when E is in some sense a category of spaces, and is even traditional in algebraic geometry, for instance when E is the dual of the category A of commutative k-algebras. The category E is in this case essentially the category of affine schemes over k. We elaborate a little on this terminology for this specific case. When a commutative k-algebra B is seen in the dual category E , it is often denoted spec B or B. If X is an object in E , the corresponding algebra is often denoted O(X ), and called the algebra of functions on X ; more precisely, the algebra of scalar valued functions on X . The category E is in this case a category equipped with a canonical commutative ring object R, namely k[X ], whose geometric meaning is that it is the number line. The ring structure of R in E comes about from the canonical co-ring structure of k[X ] in the category of commutative k-algebras. Now ring structure on the geometric line is elementary and well understood, since the time of Euclid, essentially, whereas the notion of coring is not elementary, and is a much more recent invention. This is why (E , R) is well suited to axiomatic abstraction, as in [3]. 11

The reason why homE (I, R) is a commutative ring (even a k-algebra) in E is that it is isomorphic to O(I); for, homE (I, R) ∼ = O(I), = homE (I, k[X ]) ∼ = homA (k[X ], O(I)) ∼

(15)

the last isomorphism because k[X ] is the free k-algebra in one generator X . Having a commutative ring object like R in a category E is the first necessary condition for having the wonderful tool of coordinates available for the geometry in E . Thus we have the coordinate vector spaces Rn ; in the category of affine schemes, this is k[X1, . . . , Xn ]. This object is an R-module object. The R-module structure may be described in the same way, in terms of C-parametrized points of Rn , as when we described the ring structure of R in such terms; a C-parametrized point of Rn amounts to an n-tuple of elements in the algebra C, and C-parametrized points of R amount to elements of C. So the R-module structure of Rn comes about from the C-module structure of Cn , for arbitrary C. If C is a finitely presented k-algebra, the object (space) C embeds into some n R , since a finite presentation of C, with n generators gives rise to a surjective (hence epimorphic) algebra map k[X1, . . . , Xn] → C. Since we have the notion of “neighbours” for algebra maps, we have a notion of neighbours for maps in E ; and it is preserved by pre- and post-composition. In particular, the embedding map e : C → Rn , obtained from a presentation of C with n generators, preserves the property of being neighbours, for parametrized families of points of C. The embedding also reflects the neighbour relation, in the sense that if x and y are points of C, and e(x) ∼ e(y), then x ∼ y. This is just a reformulation of (6). Note the traditional replacement of the notation e ◦ x by e(x). This is the “symbolic” counterpart of considering maps I → M as (I-parametrized) points 2 of M. Furthermore, it is tradition in algebraic geometry not always to be specific about the space I of parameters for a parametrized point I → M; thus, one talks about “points (x, y) of the unit circle S given by x2 + y2 = 1”, without explicit mention of whether it one means a real point, a rational point, a complex point, . . . ; therefore, I is omitted from notation, and one writes (x, y) ∈ S. This in particular applies when the statement or notion applies to any (parametrized) point of M, regardless of its parameter space I. 2 In

algebraic geometry, the terminology “I-valued point of M” is also used, see e.g. [1] p. 209. In [3], Part II, such a thing is called a “generalized element of M, defined at stage I”, and a more elaborate description of the ‘logic’ of generalized elements is presented.

12

An example of this usage is for the neighbour relation in affine schemes. Consider two map f and g between k-algebras B and C, as in (1). In the category of affine schemes, these are then neighbour maps f and g: C → B, i.e. neighbour points of B (parametrized by C); we write f ∈ B, g ∈ B. With this usage, Proposition 2.3 may be reformulated as Proposition 5.1 Given two points (a1 , . . ., an ) and (b1, . . . , bn ) ∈ Rn . Then they are neighbours iff (bi − ai ) · (b j − a j ) = 0 (16) for all i, j = 1, . . . , n. Here the (common) parameter space C of the ai s and bi s is not mentioned explicitly; it could be any affine scheme. Note that (16) is typographically the same as (8); in (16), ai and b j are (parametrized) points of R (parametrized by C), in (8), they are elements in the algebra C; but these data correspond, by (15), and this correspondence preserves algebraic structure. Similarly, Proposition 2.4 gets the reformulation: Proposition 5.2 A p+1-tuple {ai j } of points in Rn form an infinitesimal p-simplex iff the equations (9) hold. This formulation, as the other formulations in “synthetic” terms, are the ones that are suited to axiomatic treatment, as in Synthetic Differential Geometry, which almost exclusively3 assumes a given commutative ring object R in a category E , preferably a topos, as a basic ingredient in the axiomatics. (The category E of affine schemes is not a topos, but the category of presheaves on E is, and it, and some of its subtoposes, are the basic categories considered in modern algebraic geometry, like [2].) As a further illustration of the “synthetic” language, the algebraic formulation of the neighbour relation between algebra maps given in (3) (assuming 2 ∈ k is invertible) may be rendered: Proposition 5.3 For any affine scheme B, scalar valued functions on B detect when points x and y of B are neighbours; i.e. if α (x) ∼ α (y) for all α : B → R, then x ∼ y. 3 Exceptions are found in

[9] (where R is constructed out of an assumed infinitesimal object T ); and in [5] and[7], where part of the reasoning does not assume any algebraic notions.

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Here x and y are points of B, say parametrized by C, i.e. they are maps C → B, so they correspond to algebra maps f and g : B → C; and α : B → R corresponds to a ∈ B. The fact (6) gets the following formulation: Proposition 5.4 Given an affine scheme B. Then for any finite presentation (with n generators, say) of the algebra, the corresponding embedding e : B → Rn (preserves and) reflects the relation ∼. So for x and y points in Rn , they come via e from a pair of neighbour points in B iff they satisfy 1) the equations in n variables defining B in the presentation; and 2) satisfy the equations for being neighbours in Rn . In other words, the intrinsically defined neighbour relation on B (essentially (1)) may be described purely equationally, using an finite equational presentation of B. Or, in more elementary tems, which the synthetic tradition is very apt for utilizing: Given a finite set of equations with coefficients from R. If of x0 , . . . , x p are points in Rn and each of them satisfies the equations, then so does any affine combination of them, provided the points are mutual neighbours. But note that Proposition 5.4, together with the constructions of Sections 1 and 3 allow us to conclude that the neighbour conditions, and the point constructed by affine combinations in Rn , is intrinsic to the affine scheme B in question, and does not depend on an equational presentation of B.

References [1] L. Breen and W. Messing, Combinatorial Differential Forms, Advances in Math. 164 (2001), 203-282. [2] M. Demazure and P. Gabriel, Groupes Algébriques, Tome I, Masson & Cie/North Holland 1970. [3] A. Kock, Synthetic Differential Geometry, London Math. Soc. Lecture Notes Series 51 (1981); 2nd ed., London Math. Soc. Lecture Notes Series 333 (2006). [4] A. Kock, Differential forms as infinitesimal cochains, J. Pure Appl. Alg. 154 (2000), 257-264.

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[5] A. Kock, The osculating plane of a space curve - synthetic formulations, Rendiconti del Circolo Mat. Palermo, Serie II, Suppl. 64 (2000), 67-79. [6] A. Kock, Some matrices with nilpotent entries, and their determinants, arXiv:math.RA/0612435v1, 2006. [7] A. Kock: Envelopes - notion and definiteness, Beiträge zur Algebra und Geometrie 48 (2007), 345-350. [8] A. Kock, Synthetic Geometry of Manifolds, Cambridge Tracts in Mathematics 180, Cambridge University Press 2010. [9] F.W. Lawvere, Euler’s Continuum Functorially Vindicated, vol. 75 of The Western Ontario Series in Philosophy of Science, 2011. Anders Kock, August 2015 Dept. of Mathematics, University of Aarhus, Denmark kock (at) math.au.dk

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